Theorem ssint 4302

Description: Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52 and its converse. (Contributed by NM, 14-Oct-1999.)

Assertion

Ref	Expression
ssint

Distinct variable groups:   ,   ,

Proof of Theorem ssint

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfss3 3493	. 2
2		vex 3112	. . . 4
3	2	elint2 4293	. . 3
4	3	ralbii 2888	. 2
5		ralcom 3018	. . 3
6		dfss3 3493	. . . 4
7	6	ralbii 2888	. . 3
8	5, 7	bitr4i 252	. 2
9	1, 4, 8	3bitri 271	1

Syntax hints:  <->wb 184  e.wcel 1818  A.wral 2807  C_wss 3475  |^|cint 4286

This theorem is referenced by:  ssintab  4303  ssintub  4304  iinpw  4419  trint  4560  oneqmini  4934  fint  5769  sorpssint  6590  iscard2  8378  coftr  8674  isf32lem2  8755  inttsk  9173  isacs1i  15054  mrelatglb  15814  fbfinnfr  20342  fclscmp  20531  dfrtrcl2  29071  fneint  30166  topmeet  30182  igenval2  30463  ismrcd1  30630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-v 3111  df-in 3482  df-ss 3489  df-int 4287